We give optimal gap conditions using Lipschitz constants of the nonlinear
terms and growth bounds of the linear terms that imply the existence of infinite
dimensional Lipschitz invariant manifolds for systems of semilinear equations on
Banach spaces. This result improves and generalizes recent theorems by C. Foias
and by N. Castañeda and R. Rosa. The result is also shown to imply the existence
of invariant manifolds for nonautonomous equations and semilinear skew-product
flows. Also, generalizations for smoothness of invariant manifolds are given.